The Process In-Between

Hilhorst (1981)]. We point out also [Roynette et al. (2003)], that focuses instead on exponential perturbations of Brownian motion. Stiff in the Brownian context, [Punaki (2005)] derives scahng hmits for a case in which boundary conditions, chosen to favor paths in the upper half-plane, are in competition with a bulk potential that favors the lower half-plane. Of course between what we are doing and the papers we have just mentioned there is the fundamental difference that we are deahng only with renewal processes, and not with the returns to zero of a random walk, or another process. We stress however that for the cases we have mentioned it is not so difficult to go from a weak convergence result of the zero level set to the convergence of the full process. This is due to the fact that, conditionally on the location of the zeros, the trajectory of the process in between is just an unperturbed random walk excursion. 

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